Israel Journal of Mathematics

, Volume 62, Issue 3, pp 311–325

The linear arboricity of graphs

Article

DOI: 10.1007/BF02783300

Cite this article as:
Alon, N. Israel J. Math. (1988) 62: 311. doi:10.1007/BF02783300

Abstract

Alinear forest is a forest in which each connected component is a path. Thelinear arboricity la(G) of a graphG is the minimum number of linear forests whose union is the set of all edges ofG. Thelinear arboricity conjecture asserts that for every simple graphG with maximum degree Δ=Δ(G),\(la(G) \leqq [\frac{{\Delta + 1}}{2}].\). Although this conjecture received a considerable amount of attention, it has been proved only for Δ≦6, Δ=8 and Δ=10, and the best known general upper bound for la(G) is la(G)≦⌈3Δ/5⌉ for even Δ and la(G)≦⌈(3Δ+2)/5⌉ for odd Δ. Here we prove that for everyɛ>0 there is a Δ00(ɛ) so that la(G)≦(1/2+ɛ)Δ for everyG with maximum degree Δ≧Δ0. To do this, we first prove the conjecture for everyG with an even maximum degree Δ and withgirth g≧50Δ.

Copyright information

© Hebrew University 1988

Authors and Affiliations

  • N. Alon
    • 1
  1. 1.Department of Mathematics, Sackler Faculty of Exact SciencesTel Aviv UniversityRamat Aviv, Tel AvivIsrael